Contents

  1. 1. Introduction
  2. 2. The Octonion Algebra O
  3. 3. Lie Derivation of Octonion Algebra O
  4. 4. Characterizing Lie Derivation of Octonion Algebra O
  5. 5. Lie Centralizer of Octonion Algebra O
  6. 6. Generalized Lie Derivation of Octonion Algebra O
  7. 7. Conclusion
  8. Acknowledgements
  9. Conflict of Interest

Ars Combinatoria

  • Research article

On Lie Derivations, Generalized Lie Derivations and Lie Centralizers of Octonion Algebras

Minahal Arshad1, M. Mobeen Munir1
1Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan.
  • Received: 21/07/2023
  • Revised: 19/01/2024
  • Accepted: 16/11/2023
  • Published: 22/11/2023

Abstract

Let L be a unital ring with characteristic different from 2 and O(L) be an algebra of Octonion over L. In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on O(L). The matrix representation of Lie derivation of O(L) and its decomposition in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. The result about the decomposition of Lie centralizer of O in terms of Lie centralizer and Jordan centralizer of L is given. Moreover, the matrix representation of generalized Lie derivation (also known as D-Lie derivation) of O(L) is computed.

Keywords: The Octonion algebra, Lie Derivations, D-Lie Derivations, Lie Centralizer

1. Introduction

It is well known that derivation is a function of an algebra which generalizes certain features of the derivative operator. It gives an interesting insight to understand the structure and local properties of an algebra. The concept to study the structure of ring theory and derivations though established long back, but got stimulated after Posner in [1] described some important results on the derivations of prime rings. A fundamental problem in the theory of derivations is to determine all the derivations on an algebra. Over the years, many important variants of derivation have been presented. Among these variants, Lie derivation and its generic extension is currently more interesting and attracting. Lie centralizer is not so common but very meaningful variant. These variants are being widely discussed now a days.

Let O be a unital algebra and the center of O is denoted by Z(O). We denote the commutator (Lie product) and Jordan product of x1, x2 by [x1,x2]=x1x2−x2x1 and x1∘x2=x1x2+x2x1 respectively for all x1,x2∈O. We say that a ring O is an F-algebra (F is a field) if O is an F-vector space equipped with a bilinear product. Let D:O→O be an additive map. We say D is a derivation (respectively Jordan derivation) if D[x1,x2]=[D(x1),x2]+[x1,D(x2)] (respectively D(x2)=D(x)x+xD(x)), for all x,x1,x2∈O. For an element α∈O, the mapping Iα:O→O given by Iα(x)=xα−αx for all x∈O is called an inner derivation of O induced by α. Let ξ:O→O be an additive map. The map ξ is said to be right (left) centralizer if ξ(x1x2)=x1ξ(x2) (ξ(x1x2)=ξ(x1)x2) for all x1,x2∈O. We say that φ is an Jordan centralizer if φ(x1∘x2)=φ(x1)∘x2 for all x1,x2∈O. An additive map ξ:O→O is called a Lie centralizer if ξ[x1,x2]=[ξ(x1),x2] for all x1,x2∈O.

The characterization of Lie centralizer of quaternion algebra is given in [2]. The characterization of Lie centralizers and their generalizations is now widely studied on different kinds of algebras by many authors in ([3,4,5,6,7]). Fosner et al. characterised Lie centralizers of triangular tings and nest algebras in [3]. Ashrafi et al. computed multiplicative generalized Lie n-derivations of unital rings with idempotents and σ-centralizers generalized matrix algebras in [4] and [5] respectively. Fadaee et al. characterized Lie centralizers at the zero product of generalized matrix algebras in [6] and Lie triple centralizers of generalized matrix algebras in [7]. In [8], Martindale described the standard form of Lie derivation on certain primitive rings. Similar results have been discussed on von Neumann algebras by Miers [9]. Mokhtari et al. computed Lie derivations on trivial extension algebras in [10]. It is obvious that every Lie derivation is a generalized Lie derivation. Besides there are two different definitions of generalized Lie derivations in literature. One is introduced by Atsushi Nakajima [11] which is stated as: for an additive map F:O→O, we say that F is generalized Lie derivation (also known as D-Lie derivation) if there exist a Lie derivation D:O→O such that F[x1,x2]=[F(x1),x2]+[x1,D(x2)] for all x1,x2∈D and the other is by Bojan Hvala [12] which is stated as: for an additive map F:O→O, we say that F is generalized Lie derivation if there exist a linear map D:O→O such that F[x1,x2]=F(x1)x2−F(x2)x1+x1D(x2)−x2D(x1) for all x1,x2∈D. Both definitions are equally being discussed by mathematicians. Hvala discussed the generalized Lie derivations on rings and proved that every generalized Lie derivation on a prime ring can be written as the sum of a generalized derivation and a central map. Benkovič [13] proved that every generalized Lie derivation from a unital algebra onto a unitary bimodule can be written in the sum of a generalized derivation and a central map that vanishes on the commutators of the algebra. The description of generalized Lie derivation of Lie ideals of prime algebras and nonlinear generalized Lie derivation of some classical triangular algebras are respectively given in [14, 15]. The generalized Lie derivations of prime rings are discussed in [16] by using both definitions. Hvala’s definition of generalized Lie derivation covers both generalized derivations and D-Lie derivations. On the other hand, Nakajima’s definition is more favourable which unifies the notions of Lie derivation and Lie centralizer. We will, in particular focus on Nakajima’s definition to compute the matrix representation of generalized Lie derivations of algebra of Octonion. The discussion about local and 2-local derivation of Octonion algebra have been discussed in [17]. Later, this discussion was extended to Cayley algebras in [18].

Mainly, our focus in this article is to describe the matrix representation as well as the characterization of Lie derivation of Octonion algebras equipped with commutator product. Authors characterized in [19], the Lie triple derivations of algebra of tensor product of some algebra T and quaternion algebra. Ghahramani et al. in [20] proved results on the characterization of generalized derivation and generalized Jordan derivation of ring of quaternion and in [21] discussed the characterization of Lie derivation and its natural generic extension of quaternion ring.

This article is arranged in the following order: Section 2 contains some minor details of Octonion algebras equipped with commutator product denoted by O. In Sections 3 and 4, matrix representation of Lie derivation as well as decomposition of Lie derivation of octonion algebra in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. Section 5 contains the characterization of Lie centralizer of Octonion algebras. In Section 6, the matrix representation of generalized Lie derivation is computed.

2. The Octonion Algebra O

Let L be an arbitrary 2-torsion free unital ring. The octonion algebra (denoted by O) over L is a class of non-associative algebra. It is a unital nonassociative algebra of dimension 8 with the basis B={e0,e1,e2,e3,e4,e5,e6,e7} and the product defined in the following table.

ej
ei ei.ej e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 e1 e2 e3 e4 e5 e6 e7
e1 e1 −e0 e3 −e2 e5 −e4 −e7 e6
e2 e2 −e3 −e0 e1 e6 e7 −e4 −e5
e3 e3 e2 −e1 −e0 e7 −e6 e5 −e4
e4 e4 −e5 −e6 −e7 −e0 e1 e2 e3
e5 e5 e4 −e7 e6 −e1 −e0 −e3 e2
e6 e6 e7 e4 −e5 −e2 e3 −e0 −e1
e7 e7 −e6 e5 e4 −e3 −e2 e1 −e0

<span id="corank1" label="corank1"

The table can be summarized as follows: ei.ej={ej,if i=0;ei,if j=0;−δije0+ϵijkek,otherwise, where δij is the Kronecker delta and ϵijk is a completely antisymmetric tensor with value +1 when ijk=123, 145, 176, 246, 257, 347, 365.
An octonion x is of the form x=x0e0+x1e1+x2e2+x3e3+x4e4+x5e5+x6e6+x7e7 with real coefficients xi. By using the product defined in the table given above, we can have the following relations; [e1,e2]=2e3,[e1,e3]=−2e2,[e1,e4]=2e5,[e1,e5]=−2e4,[e1,e6]=−2e7,[e1,e7]=2e6,[e2,e3]=2e1,[e2,e4]=2e6,[e2,e5]=2e7,[e2,e6]=−2e4,[e2,e7]=−2e5,[e3,e4]=2e7,[e3,e5]=−2e6,[e3,e6]=2e5,[e3,e7]=−2e4,[e4,e5]=2e1,[e4,e6]=2e2,[e4,e7]=2e3,[e5,e6]=−2e3,[e5,e7]=2e2,[e6,e7]=−2e1. Using the above product on the basis as Lie product and extend it by linearity, we can equip this product on O.

3. Lie Derivation of Octonion Algebra O

In this section, we compute matrix representation of Lie derivation of the octonion Algebra. Let D:O→O be a Lie derivation. D admits a matrix representation with respect to the basis, which is an 8×8 matrix [D]=(βij)T whose entries are defined by (1)D(ei−1)=∑j=18βijej−1,1≤i≤8. Each column of [D] is an element of O.

Theorem 1. The algebra of Lie derivations of Octonions is generated by the following matrices: =(β11000000000−β23−β24−β25−β26−β27−β280β230−β34−β35−β36−β37−β380β24β340−β27+β36−β28−β35β25−β38β26+β370β25β35β27−β360−β56−β57−β580β26β36β28+β35β560−β23−β58−β24+β570β27β37−β25+β38β57β23+β580−β34−β560β28β38−β26−β37β58β24−β57β34+β560)

Proof. Let D be a Lie derivation of O. Then we write D(ei−1)=∑j=18βijej−1,1≤i≤8 for some arbitrary βij′s∈L. Applying D on the identity [e0,ej]=0 for 1≤j≤7. So, we get β1j=0 for 2≤j≤7.
Applying D on identities e3=12[e1,e2], e3=12[e4,e7] and e3=12[e6,e5], we get D(e3)=12(2β22e3−2β24e1−2β25e6−2β26e7+2β27e4+2β28e5+2β33e3−2β34e2+2β35e5−2β36e4−2β37e7+2β38e6)=(2β52e6−2β53e5−2β54e4+2β55e3+2β56e2−2β57e1−2β82e5−2β83e6−2β84e7+2β86e1+2β87e2+2β88e3)=(−2β62e7−2β63e4+2β64e5+2β65e2−2β66e3+2β68e1+2β72e4−2β73e7+2β74e6−2β75e1−2β77e3+2β78e2).

Now by applying D on e1=12[e2,e3], e1=12[e4,e5] and e1=−12[e6,e7], we get D(e1)=12(−2β32e2+2β33e1−2β35e7+2β36e6−2β37e5+2β38e4−2β42e3+2β44e1+2β45e6+2β46e7−2β47e4−2β48e5)=12(−2β52e4+2β53e7−2β54e6+2β55e1+2β57e3−2β58e2−2β62e5−2β63e6−2β64e7+2β66e1+2β67e2+2β68e3)=−12(2β72e6−2β73e5−2β74e4+2β75e3+2β76e2−2β77e1+2β82e7+2β83e4−2β84e5−2β85e2+2β86e3−2β88e1). Now by applying D on [e1,e3], [e4,e6] and [e5,e7], we get D(e2)=−12(−2β22e2+2β23e1−2β25e7+2β26e6−2β27e5+2β28e4+2β43e3−2β44e2+2β45e5−2β46e4−2β47e7+2β48e6)=12(−2β52e7−2β53e4+2β54e5+2β55e2−2β56e3+2β58e1−2β72e5−2β73e6−2β74e7+2β76e1+2β77e2+2β78e3)=12(2β62e6−2β63e5−2β64e4+2β65e3+2β66e2−2β67e1+2β82e4−2β83e7+2β84e6−2β85e1−2β87e3+2β88e2). Similarly by applying D on all the remaining identities, we get for D(e4) D(e4)=−12(−2β22e4+2β23e7−2β24e6+2β25e1+2β27e3−2β28e2+2β63e3−2β64e2+2β65e5−2β66e4−2β67e7+2β68e6)=−12(−2β32e7−2β33e4+2β34e5+2β35e2−2β36e3+2β38e1−2β72e3+2β74e1+2β75e6+2β76e7−2β77e4−2β78e5)=−12(2β42e6−2β43e5−2β44e4+2β45e3+2β46e2−2β47e1+2β82e2−2β83e1+2β85e7−2β86e6+2β87e5−2β88e4). For D(e5), D(e5)=12(2β22e5+2β23e6+2β24e7−2β26e1−2β27e2−2β28e3+2β53e3−2β54e2+2β55e5−2β56e4−2β57e7+2β58e6)=−12(2β32e6−2β33e5−2β34e4+2β35e3+2β36e2−2β37e1−2β82e3+2β84e1+2β85e6+2β86e7−2β87e4−2β88e5)=12(−2β42e7−2β43e4+2β44e5+2β45e2−2β46e3+2β48e1+2β72e2−2β73e1+2β75e7−2β76e6+2β77e5−2β78e4). For D(e6), D(e6)=12(2β22e6−2β23e5−2β24e4+2β25e3+2β26e2−2β27e1+2β83e3−2β84e2+2β85e5−2β86e4−2β87e7+2β88e6)=12(2β32e5+2β33e6+2β34e7−2β36e1−2β37e2−2β38e3−2β52e3+2β54e1+2β55e6+2β56e7−2β57e4−2β58e5)=−12(−2β42e4+2β43e7−2β44e6+2β45e1+2β47e3−2β48e2+2β62e2−2β63e1+2β65e7−2β66e6+2β67e5−2β68e4). For D(e7), D(e7)=−12(−2β22e7−2β23e4+2β24e5+2β25e2−2β26e3+2β28e1+2β73e3−2β74e2+2β75e5−2β76e4−2β77e7+2β78e6)=12(−2β32e4+2β33e7−2β34e6+2β35e1+2β37e3−2β38e2−2β62e3+2β64e1+2β65e6+2β66e7−2β67e4−2β68e5)=12(2β42e5+2β43e6+2β44e7−2β46e1−2β47e2−2β48e3+2β52e2−2β53e1+2β55e7−2β56e6+2β57e5−2β58e4). By comparing the coefficients, we get βi1=0 for 2≤i≤8, βij=0 for 2≤i,j≤8 with i=j and βji=−βij for 2≤i,j≤8 with i≠j. Specifically, β45=β27−β36, β46=β28+β35, β47=−β25+β38, β48=−β26−β37, β67=β23+β58, β68=β24−β57 and β78=β34+β56. ◻

Theorem 2. Let x=∑i=18xiei−1∈O. Let D:O→O be a Lie derivation. Then D(x) can be written as D(x)=x1β11e0+(−x3β23−x4β24−x5β25−x6β26−x7β27−x8β28)e1+(x2β23−x4β34−x5β35−x6β36−x7β37−x8β38)e2+(x2β24+x3β34+x5(−β27+β36)+x6(−β28−β35)+x7(β25−β38)+x8(β26+β37))e3+(x2β25+x3β35+x4(β27−β36)−x6β56−x7β57−x8β58)e4+(x2β26+x3β36+x4(β28+β35)+x5β56+x7(−β23−β58)+x8(−β24+β57))e5+(x2β27+x3β37+x4(−β25+β38)+x5β57+x6(β23+β58)+x8(−β34−β56))e6+(x2β28+x3β38+x4(−β26−β37)+x5β58+x6(β24−β57)+x7(β34+β56))e7.

Proof. Applying D on x gives D(x)=∑i=18xiD(ei−1). Substituting the values of D(ei)′s, which is computed in matrix representation of D in above theorem yields D(x)=x1β11e0+x2(β23e2+β24e3+β25e4+β26e5+β27e6+β28e7)+x3(−β23e1+β34e3+β35e4+β36e5+β37e6+β38e7)+x4(−β24e1−β34e2+(β27−β36)e4+(β28+β35)e5+(−β25+β38)e6+(−β26−β37)e7)+x5(−β25e1−β35e2+(−β27+β36)e3+β56e5+β57e6+β58e7)+x6(−β26e1−β36e2+(−β28−β35)e3−β56e4+(β23+β58)e6+(β24−β57)e7)+x7(−β27e1−β37e2+(β25−β38)e3−β57e4+(−β23−β58)e5+(β34+β56)e7)+x8(−β28e1−β38e2+(β26+β37)e3−β58e4+(−β24+β57)e5+(−β34−β56)e6). Summarizing the above expression yields our required result. ◻

4. Characterizing Lie Derivation of Octonion Algebra O

Our next task is to present characterization of Lie derivations of the algebra of Octonion. In Theorem 2.2 of [21], it is shown that if S be a 2-torsion free ring and R=H(S) be quaternion ring, then every Lie derivation of R can be decomposed in terms of Jordan derivation and Lie derivation of S and an inner derivation of R, for every element t∈R. Here, we have:

Theorem 3. Let D:O→O be a Lie derivation. Then there exist an element A in O, a Lie derivation δ and a Jordan derivation ψ on L such that D(t)=δ(x1)e0+∑i=28ψ(xi)ei−1+IA(t) for every element t=∑i=18xiei−1∈O.

Proof. Since D is an additive map, we can write (2)D(ei−1)=∑j=18βijej−1,1≤i≤8. for some βij′s∈L. It can be easily seen that β11∈Z(L). Next, we will fine D(sei)′s with i=0,1,…,7, for arbitrary l∈L. Set D(le1)=∑i=18xiei−1. Applying D on [le1,e1], we get 0=D[le1,e1]=−2x3e3+2x4e2−2x5e5+2x6e4+2x7e7−2x8e6+(l∘β23)e3−(l∘β24)e2+(l∘β25)e5−(l∘β26)e4−(l∘β27)e7+(l∘β28)e6. By comparing the coefficients, we get x3=12(l∘β23),x4=12(l∘β24),x5=12(l∘β25),x6=12(l∘β26),x7=12(l∘β27),x8=12(l∘β28), which implies D(le1)=x1e0+x2e1+12(l∘β23)e2+12(l∘β24e3+12(l∘β25)e4+12(l∘β26)e5+12(l∘β27)e6+12(l∘β28)e7. Now, applying D on the identities le3=12[le1,e2], le2=−12[le1,e3], le1=12[le2,e3], le4=−12[le1,e5], le5=12[le1,e4], le6=12[le1,e7], le7=−12[le1,e6] and putting x2=ψ(l) where ψ:L→L is an additive map which is uniquely determined by D, we get D(le1)=12Iβ34(l)e0+ψ(l)e1+12(l∘β23)e2+12(l∘β24e3+12(l∘β25)e4+12(l∘β26)e5+12(l∘β27)e6+12(l∘β28)e7.D(le2)=−12Iβ24(l)e0−12(l∘β23)e1+ψ(l)e2+12(l∘β34)e3+12(l∘β35)e4+12(l∘β36)e5+12(l∘β37)e6+12(l∘β38)e7.D(le3)=12Iβ23(l)e0−12(l∘β24)e1−12(l∘β34)e2+ψ(l)e3+12(l∘(β27−β36))e4+12(l∘(β28+β35))e5−12(l∘(β25−β38))e6−12(l∘(β26+β37))e7D(le4)=−12Iβ26(l)e0−12(l∘β25)e1−12(l∘β25)e2−12(l∘(β27−β36))e3(3)+ψ(l)e4+12(l∘β56)e5+12(l∘β57)e6+12(l∘β58)e7D(le5)=12Iβ25(l)e0−12(l∘β26)e1−12(l∘β36)e2−12(l∘(β28+β35))e3−12(l∘β56)e4+ψ(l)e5+12(l∘(β23+β58))e6+12(l∘(β24−β57))D(le6)=12Iβ28(l)e0−12(l∘β27)e1−12(l∘β37)e2+12(l∘(β25−β38))e3−12(l∘β57)e4−12(l∘(β23+β58))e5+ψ(l)e6+12(l∘(β34+β56))D(le7)=−12Iβ27(l)e0−12(l∘β28)e1−12(l∘β38)e2+12(l∘(β26+β37))e3−12(l∘β58)e4−12(l∘(β24−β57))e5−12(l∘(β34+β56))e6+ψ(l)e7. Next, let l∈L be arbitrary and put D(le0)=D(l)=x1e0+x2e1+x3e2+x4e3+x5e4+x6e5+x7e6+x8e7. Applying D on [le0,e1]=0 and using (2), we obtain 0=D[le0,e1]=−2x3e3+2x4e2−2x5e4+2x6e4+2x7e7−2x8e6+Iβ23(l)e2+Iβ24(l)e3+Iβ25(l)e4+Iβ26(l)e5+Iβ27(l)e6+Iβ28(l)e7. By comparing the coefficients, we get x3=12Iβ24(l),x4=−12Iβ23(l),x5=12Iβ26(l),x6=−12Iβ25(l),x7=−12Iβ28(l),x8=12Iβ27(l). Applying D on the identities [le0,ei] where i=2,…,7 and taking x1=δ(l) for some additive map δ:L→L uniquely determined by D, we get (4)D(le0)=δ(l)e0−12Iβ34(l)e1+12Iβ24(l)e2−12Iβ23(l)e3+12Iβ26(l)e4−12Iβ25(l)e5−12Iβ28(l)e6+12Iβ27(l)e7 and −Iβ34=−Iβ56=Iβ34+β56Iβ24=−Iβ57=−Iβ24−β57(5)−Iβ23=−Iβ58=Iβ23+β58Iβ26=Iβ37=−Iβ26+β37−Iβ25=Iβ38=−I−β25+β38−Iβ28=−Iβ35=Iβ28+β35Iβ27=−Iβ36=−Iβ27−β36. Replacing l by [l1,l2] in (4), for some l1,l2∈L, we infer that δ is a Lie derivation of L. Moreover, applying D on the identity [l1e1,l2e2]=(l1∘l2)e3 and using the foregoing calculations, we can see that ψ is a Jordan derivation. Now let t=∑i=18xiei−1∈O be an arbitrary element. Using (???), (4) and (???), we find that D(t)=δ(x1)e0+∑i=28ψ(xi)ei−1+h(t) where h(t)=e02(Iβ34(x2)−Iβ24(x3)+Iβ23(x4)−Iβ26(x5)+Iβ25(x6)+Iβ28(x7)−Iβ27(x8))+e12(−Iβ34(x1)−(x3∘β23)−(x4∘β24)−(x5∘β25)−(x6∘β26)−(x7∘β27)−(x8∘β28))+e22(Iβ24x1+(x2∘β23)−(x4∘β34)−(x5∘β35)−(x6∘β36)−(x7∘β37)−(x8∘β38))+e32(−Iβ23x1+(x2∘β24)+(x3∘β34)−(x5∘(β27−β36))−(x6∘(β28+β35))+(x7∘(β25−β38))+(x8∘(β26+β37)))+e42(Iβ26x1+(x2∘β25)+(x3∘β35)+(x4∘(β27−β36))−(x6∘β56)−(x7∘β57)−(x8∘β58))+e52(−Iβ25(x1)+(x2∘β26)+(x3∘β36)+(x4∘(β28+β35))+(x5∘β56)−(x7∘(β23+β58))−(x8∘(β24−β57)))+e62(−Iβ28x1+(x2∘β27)+(x3∘β37)−(x4∘(β25−β38))+(x5∘β57)+(x6∘(β23+β58))−(x8∘(β34+β56)))+e72(Iβ27x1+(x2∘β28)+(x3∘β38)−(x4∘(β26+β37))+(x5∘β58)+(x6∘(β24−β57))+(x7∘(β34∘+β56))). It can be easily verified that h(t)=IA(t) where t=∑i=18xiei−1 and A=12(−β34e1+β24e2−β23e3+β26e4−β25e5−β28e6+β27e7). Consequently, D(t)=δ(x1)e0+∑i=28ψ(xi)ei−1+IA(t). ◻

5. Lie Centralizer of Octonion Algebra O

This section contains the characterization of Lie centralizer of octonion algebra. In Theorem 2.1 of [2], it is shown that if S is a 2-torsion free unital ring and R=H(S) is quaternion ring. Then every Lie centralizer of R can be represented in terms of a Lie centralizer and Jordan centralizer of S. Here, we have:

Theorem 4. Let ξ:O→O be a Lie centralizer. Then there exists a Lie centralizer α and a Jordan centralizer φ on L such that ξ(t)=α(x1)e0+∑i=28φ(xi)ei−1 for every element t=∑i=18xiei−1∈O.

Proof. We have already assumed the form ξ(ei−1)=∑j=18βijej−1,1≤i≤8 for some βij′s∈L. Since ξ is a Lie centralizer, we have ξ(e3)=12ξ[e1,e2]=12[ξ(e1),e2]=12(2β22e3−2β24e1−2β25e6−2β26e7+2β27e4+2β28e5)=−β24e1+β22e3+β27e4+β28e5−β25e6−β26e7. Furthermore, ξ(e1)=12ξ[e2,e3]=12[ξ(e2),e3]=β33e1−β32e2+β38e4−β37e5+β36e6−β35e7. By comparing the coefficients, we have β21=β24=β42=β41=β43=0, β22=β33=β44, β25=β38=−β47, β26=−β37=−β48, β27=β36=β45, β28=−β35=β46, which reduces ξ(e1) and ξ(e3) to ξ(e1)=β22e1+β23e2+β25e4+β26e5+β27e6+β28e7ξ(e3)=β22e3+β27e4+β28e5−β25e6−β26e7. Applying ξ on the identities e2=12[e3,e1], e4=12[e5,e1], e5=12[e1,e4], e6=12[e1,e7], e7=12[e6,e1], we get ξ(e2)=β22e2+β28e4−β27e5+β26e6−β25e7ξ(e4)=β28e2+β27e3+β22e4+β23e7ξ(e5)=−β26e1−β27e2+β28e3+β22e5+β23e6ξ(e6)=−β27e1+β26e2+β25e3−β23e5+β22e6ξ(e7)=β25e2−β26e3−β23e4+β22e7. Now assume that, ξ(e0)=t=∑i=18xiei−1. We have 0=ξ[e0,t]=te0−e0t=−2x3e3+2x4e2−2x5e5+2x6e4+2x7e7−2x8e6, which implies x3=x4=x5=x6=x7=x8=0. Application of ξ on the identity [e0,e2] gives x2=0, which implies ξ(e0)=x1e0=x1∈L. Let s∈L be an arbitrary then 0=ξ[e0,se1]=[ξ(e0),se1]=(x1s−sx1). From this we get, x1=ξ(e0)∈Z(L).
Let l∈L and set ξ(le1)=∑i=18xiei−1. Applying ξ on [le1,e1]=0, we get x3=x4=x5=x6=x7=x8=0, which reduces ξ(le1) to ξ(le1)=x1e0+x2e1. Now applying ξ on the identities le3=12[le1,e2], le2=12[le3,e1], le1=12[le2,e3], le5=12[le1,e4], le4=12[le5,e1], le6=12[le1,e7], le7=12[le6,e1] and taking x2=φ(l), where φ:L→L is an additive map which is uniquely determined by ξ, we get ξ(le1)=φ(l)e1,ξ(le2)=φ(l)e2,ξ(le3)=φ(l)e3,(6)ξ(le4)=φ(l)e4,ξ(le5)=φ(l)e5,ξ(le6)=φ(l)e6,ξ(le7)=φ(l)e7 Our next goal is to calculate ξ(le0) for arbitrary l∈L. Set ξ(l)=∑i=18xiei−1. Applying ξ on [l,e1]=0 and [l,e2]=0 and putting x1=α(l), where α:L→L is an additive map which is uniquely determined by ξ, we get (7)ξ(l)=α(l). Since ξ is a Lie centralizer, (7) implies that α is a Lie centralizer on L. Let l1,l2∈L. Applying ξ on the identity [l1e1,l2e2]=(l1∘l2)e3 and using (???) and (7), we get φ(l1∘l2)=φ(l1)∘l2 shows that φ is a Jordan centralizer on L. Now let t=∑i=18xiei−1 be an arbitrary element in L. By (???) and (7), we get ξ(t)=α(x1)e0+∑i=28φ(xi)ei−1, which completes the proof. ◻

6. Generalized Lie Derivation of Octonion Algebra O

Generalized derivation is an extension of natural derivation. It has many applications in the literature since it is quite helpful in the geometric classification of rings and algebras. In this section, we compute the matrix representation of generalized Lie derivation of the octonion algebra. Let F:O→O be a generalized Lie derivation. F admits a matrix representation with respect to the basis, which is an 8×8 matrix [F]=(γij)T whose entries are defined by (8)F(ei−1)=∑j=18γijej−1,1≤i≤8. Each column of [F] is an element of O.

Theorem 5. Let F:O→O be a generalized Lie derivation of O and B be the basis of O. Then the matrix representation of F is as follows =(γ11−β1100000000γ2200000000γ2200000000γ2200000000γ2200000000γ2200000000γ2200000000γ22)+(β11000000000−β23−β24−β25−β26−β27−β280β230−β34−β35−β36−β37−β380β24β340−β27+β36−β28−β35β25−β38β26+β370β25β35β27−β360−β56−β57−β580β26β36β28+β35β560−β23−β58−β24+β570β27β37−β25+β38β57β23+β580−β34−β560β28β38−β26−β37β58β24−β57β34+β560).

Proof. Let F be a generalized Lie derivation of O, then (9)F[ei,ej]=[F(ei),ej]+[ei,D(ej)] where D is the derivation of the octonion algebra.
Put i=0, then F[e0,ej]=[F(e0),ej]+[e0,D(ej)]=0 for 1≤j≤7. So, by using the equation(8), we get γ1j=0 for 2≤j≤7.
Unlike the procedure of finding the Lie derivation of O, we don’t need to verify equation (9) for the products between the octonionic units to compute the matrix representation of F. Hence generalized Lie derivation is much easier to compute once a Lie derivation is obtained.
Suppose that F(ei−1)=∑j=18γijej−1,1≤i≤8. Applying F on the identity [ei,ej] for i=j and comparing the coefficients, we get γij=βij for 2≤i,j≤8 with i≠j.

By using the same technique proposed in the previous theorem, we get, for F(e1) F(e1)=12(−2γ32e2+2γ33e1−2γ35e7+2γ36e6−2γ37e5+2γ38e4−2β42e3+2β44e1+2β45e6+2β46e7−2β47e4−2β48e5)=12(−2γ52e4+2γ53e7−2γ54e6+2γ55e1+2γ57e3−2γ58e2−2β62e5−2β63e6−2β64e7+2β66e1+2β67e2+2β68e3)=−12(2γ72e6−2γ73e5−2γ74e4+2γ75e3+2γ76e2−2γ77e1+2β82e7+2β83e4−2β84e5−2β85e2+2β86e3−2β88e1). For F(e2) F(e2)=−12(−2γ22e2+2γ23e1−2γ25e7+2γ26e6−2γ27e5+2γ28e4+2β43e3−2β44e2+2β45e5−2β46e4−2β47e7+2β48e6)=12(−2γ52e7−2γ53e4+2γ54e5+2γ55e2−2γ56e3+2γ58e1−2β72e5−2β73e6−2β74e7+2β76e1+2β77e2+2β78e3)=12(2γ62e6−2γ63e5−2γ64e4+2γ65e3+2γ66e2−2γ67e1+2β82e4−2β83e7+2β84e6−2β85e1−2β87e3+2β88e2). For F(e3) F(e3)=12(2γ22e3−2γ24e1−2γ25e6−2γ26e7+2γ27e4+2γ28e5+2β33e3−2β34e2+2β35e5−2β36e4−2β37e7+2β38e6)=12(2γ52e6−2γ53e5−2γ54e4+2γ55e3+2γ56e2−2γ57e1−2β82e5−2β83e6−2β84e7+2β86e1+2β87e2+2β88e3)=−12(−2γ62e7−2γ63e4+2γ64e5+2γ65e2−2γ66e3+2γ68e1+2β72e4−2β73e7+2β74e6−2β75e1−2β77e3+2β78e2). For F(e4) F(e4)=−12(−2γ22e4+2γ23e7−2γ24e6+2γ25e1+2γ27e3−2γ28e2+2β63e3−2β64e2+2β65e5−2β66e4−2β67e7+2β68e6)=−12(−2γ32e7−2γ33e4+2γ34e5+2γ35e2−2γ36e3+2γ38e1−2β72e3+2β74e1+2β75e6+2β76e7−2β77e4−2β78e5)=−12(2γ42e6−2γ43e5−2γ44e4+2γ45e3+2γ46e2−2γ47e1+2β82e2−2β83e1+2β85e7−2β86e6+2β87e5−2β88e4). For F(e5) F(e5)=12(2γ22e5+2γ23e6+2γ24e7−2γ26e1−2γ27e2−2γ28e3+2β53e3−2β54e2+2β55e5−2β56e4−2β57e7+2β58e6)=−12(2γ32e6−2γ33e5−2γ34e4+2γ35e3+2γ36e2−2γ37e1−2β82e3+2β84e1+2β85e6+2β86e7−2β87e4−2β88e5)=12(−2γ42e7−2γ43e4+2γ44e5+2γ45e2−2γ46e3+2γ48e1+2β72e2−2β73e1+2β75e7−2β76e6+2β77e5−2β78e4). For F(e6) F(e6)=12(2γ22e6−2γ23e5−2γ24e4+2γ25e3+2γ26e2−2γ27e1+2β83e3−2β84e2+2β85e5−2β86e4−2β87e7+2β88e6)=12(2γ32e5+2γ33e6+2γ34e7−2γ36e1−2γ37e2−2γ38e3−2β52e3+2β54e1+2β55e6+2β56e7−2β57e4−2β58e5)=−12(−2γ42e4+2γ43e7−2γ44e6+2γ45e1+2γ47e3−2γ48e2+2β62e2−2β63e1+2β65e7−2β66e6+2β67e5−2β68e4). For F(e7) F(e7)=−12(−2γ22e7−2γ23e4+2γ24e5+2γ25e2−2γ26e3+2γ28e1+2β73e3−2β74e2+2β75e5−2β76e4−2β77e7+2β78e6)=12(−2γ32e4+2γ33e7−2γ34e6+2γ35e1+2γ37e3−2γ38e2−2β62e3+2β64e1+2β65e6+2β66e7−2β67e4−2β68e5)=12(2γ42e5+2γ43e6+2γ44e7−2γ46e1−2γ47e2−2γ48e3+2β52e2−2β53e1+2β55e7−2β56e6+2β57e5−2β58e4). By comparing the coefficients, we get γi1=0 for 2≤i≤8 and γ22=γ33=γ44=γ55=γ66=γ77=γ88. ◻

Theorem 6. Let x=x1e0+x2e1+x3e2+x4e3+x5e4+x6e5+x7e6+x8e7∈O. Let F:O→O be a generalized Lie derivation with respect to D then F(x) can be written as F(x)=x1γ11e0+(x2γ22−x3β23−x4β24−x5β25−x6β26−x7β27−x8β28)e1+(x2β23+x3γ22−x4β34−x5β35−x6β36−x7β37−x8β38)e2+(x2β24+x3β34+x4γ22+x5(−β27+β36)+x6(−β28−β35)+x7(β25−β38)+x8(β26+β37))e3+(x2β25+x3β35+x4(β27−β36)+x5γ22−x6β56−x7β57−x8β58)e4+(x2β26+x3β36+x4(β28+β35)+x5β56+x6γ22+x7(−β23−β58)+x8(−β24+β57))e5+(x2β27+x3β37+x4(−β25+β38)+x5β57+x6(β23+β58)+x7γ22+x8(−β34−β56))e6+(x2β28+x3β38+x4(−β26−β37)+x5β58+x6(β24−β57)+x7(β34+β56)+x8γ22)e7.

Proof. Applying F on x gives F(x)=x1F(e0)+x2F(e1)+x3F(e2)+x4F(e3)+x5F(e4)+x6F(e5)+x7F(e6)+x8F(e7). Substituting the values of F(ei)′s, which is computed in matrix representation of F in above theorem yields F(x)=x1γ11e0+x2(γ22e1+β23e2+β24e3+β25e4+β26e5+β27e6+β28e7)+x3(−β23e1+γ22e2+β34e3+β35e4+β36e5+β37e6+β38e7)+x4(−β24e1−β34e2+γ22e3+(β27−β36)e4+(β28+β35)e5+(−β25+β38)e6+(−β26−β37)e7)+x5(−β25e1−β35e2+(−β27+β36)e3+γ22e4+β56e5+β57e6+β58e7)+x6(−β26e1−β36e2+(−β28−β35)e3−β56e4+γ22e5+(β23+β58)e6+(β24−β57)e7)+x7(−β27e1−β37e2+(β25−β38)e3−β57e4+(−β23−β58)e5+γ22e6+(β34+β56)e7)+x8(−β28e1−β38e2+(β26+β37)e3−β58e4+(−β24+β57)e5+(−β34−β56)e6+γ22e7). Summarizing the above expression yields our required result. ◻

Example 1. Let an arbitrary element x=∑i=18xiei−1∈O. Let β23=1, β26=1, β37=1, β58=1 and βij=0 otherwise in Lie derivation of O then D will be (10)D(x)=(−x3−x6)e1+(x2−x7)e2+2x8e3−x8e4+(x2−2x7)e5+(x3+2x6)e6+(−2x4+x5)e7. Select γ11=1 and γ22=1 in the generalized Lie derivation, then (11)F(x)=x1e0+(x2−x3−x6)e1+(x2+x3−x7)e2+(x4+2x8)e3+(x5−x8)e4+(x2+x6−2x7)e5+(x3+2x6+x7)e6+(−2x4+x5+x8)e7. Put i=1, j=7 in F[ei,ej]=[F(ei),ej]+[ei,D(ej)], we get F[e1,e7]=[F(e1),e7]+[e1,D(e7)]. The left hand side of the above equation will be F[e1,e7]=2F(e6)=−2e2−4e5+2e6. By using (11), we get F(e1)=e1+e2+e5 and using (10), we get D(e7)=2e3−e4.
Then by direct calculation the right hand side will be 2e6−4e5−2e2.

7. Conclusion

Lie algebras of derivations and its variants explore the nature of given algebras. In this research article, we have described of matrix representation as well as the characterization of Lie derivation of Octonion algebra. The characterization of Lie centralizer of Octonion algebra is also presented. We have also computed the matrix representation of generalized Lie derivations of Octonion algebras.

Acknowledgements

This research has been supported by HEC of Pakistan under the project NRPU 13150. Authors are thankful to HEC of Pakistan for supporting and funding this research.

Conflict of Interest

The authors declare no conflict of interests.

References:

  1. Posner, E.C., 1957. Derivations in prime rings. Proceedings of the American Mathematical Society, 8(6), pp.1093-1100.[Google Scholor]
  2. Bahmani, M. A. and Ghomananji, F., 2022. On the Lie centralizers of quaternion rings. Kragujevac Journal of Mathematics, 46(3), PP.471-475.[Google Scholor]
  3. Fo\v{s}ner, A. and Jing, W., 2019. Lie centralizers on triangular rings and nest algebras. Advances in Operator Theory, 4(2), PP.342-350.[Google Scholor]
  4. Ashraf, M. and Ansari, M. A., 2022. Multiplicative generalized Lie n-derivations of unital rings with idempotents. RRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 116(2), article, 92.[Google Scholor]
  5. Ashraf, M. and Ansari, M. A., 2023. σ-centralizers of generalized matrix algebras. Miskolc Mathematical Notes, 24(2), PP. 579-595.[Google Scholor]
  6. Fadaee, B. and Ghahramani, H., 2022. Lie centralizers at the zero products on generalized matrix algebras. Journal of Algebra \& Its Applications, 21(8), PP.1-22.[Google Scholor]
  7. Fadaee, B., Ghahramani, H. and Jing, W., 2023. Lie triple centralizers on generalized matrix algebras. Quaestiones Mathematicae, 46(2), pp.281-300.[Google Scholor]
  8. Martindale III, W. S., 1964. Lie derivations of primitive rings. Michigan Mathematical Journal, 11(2), pp.183-187.[Google Scholor]
  9. Miers, C. R., 1973. Lie derivations of von Neumann algebras. Duke Mathematical Journal, 40(2), pp.403-409.[Google Scholor]
  10. Mokhtari, A.H., Moafian, F. and Ebrahimi Vishki, H.R., 2017. Lie derivations on trivial extension algebras. Annales Mathematicae Silesianae, 31, pp.141-153.[Google Scholor]
  11. Nakajima, A., 2000. On generalized higher derivations. Turkish Journal of Mathematics, 24(3), pp.295-311.[Google Scholor]
  12. Hvala, B., 2007. Generalized Lie derivations in prime rings. Taiwanese Journal of Mathematics, 11(5), pp.1425-1430.[Google Scholor]
  13. Benkovi\v{c}, D., 2011. Generalized Lie derivations on triangular algebras. Linear algebra and its applications, 434(6), pp. 1532-1544.[Google Scholor]
  14. Liao, P. B. and Liu, C. K., 2009. On generalized Lie derivations of Lie ideals of prime algebras. Linear algebra and its applications, 430(4), 1236-1242. [Google Scholor]
  15. Fei, X. and Zhang, J., 2017. Nonlinear generalized Lie derivations on triangular algebras. Linear and Multilinear Algebra, 65(6), pp.1158-1170. [Google Scholor]
  16. Yarbil, N. B. and Argaç, N., 2017. A note on generalized Lie derivations of prime rings. Frontiers of Mathematics in China, 12(1), pp.247-260.[Google Scholor]
  17. Arzikulov, F. N., Karimjanov, I. A. and Uguz, S., 2020. Local and 2-local derivations of the octonion algebra. arXiv preprint arXiv:2010.16202.[Google Scholor]
  18. Ayupov, S., Elduque, A., and Kudaybergenov, K., 2021. Local and 2-local derivations of Cayley algebras. arXiv preprint arXiv:2105.08423.[Google Scholor]
  19. Asif, S., Wu, Z. and Munir, M., 2021. On the Lie triple derivations. Linear and Multilinear Algebra, pp.1-12.
  20. Ghahramani, H., Ghosseiriand, M. N. and Zadeh, L. H., 2021. Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings. Iranian Journal of Science and Technology, Transactions A: Science, 45(1), pp.305-308.[Google Scholor]
  21. Ghahramani, H., Ghosseiri, M. N. and Heidari Zadeh, L., 2019. On the Lie derivations and generalized Lie derivations of quaternion rings. Communications in Algebra, 47(3), pp.1215-1221.[Google Scholor]
Information
Guidelines
CP Initiatives
Follow CP
Facebook X-twitter Linkedin

1970-2025 CP (Manitoba, Canada) unless otherwise stated.